We started this series by introducing some indicators based on price. Our goal is to use indicators, along with price and volume, to make investment decisions: to choose when to buy or sell a financial asset. There are different ways we can incorporate price, volume, and indicators in our investment decision process. The first, the most traditional, is to interpret their patterns in a discretional way, as followers of Technical Analysis do. Indicators can also be employed in a more quantitative approach as building blocks of a trading system that removes human discretion from the investment process. Algorithmic Trading, in particular, is an approach based on trading strategies that take positions in financial instruments on their own without human intervention. We can also use price, volume, and indicators as part of a more complex machine learning model for our investment decisions.

(This post was originally published on Towards Data Science on October 7, 2020)

Whatever way we choose to use our indicators, we need to answer an important question: how good are our indicators, or combinations of indicators, to inform our investment decisions? In other words, will the use of any indicator lead to better results than not using them at all?

A process that can help us to answer this question is known as backtesting. With backtesting, we apply a trading or investment strategy to historical data to generate hypothetical results. We can then analyze those results to evaluate the profitability and the risk of our strategy.

This process has its own pitfalls: there is no guarantee that a strategy that performed well on historical data will perform well in real trading. Real trading involves many factors that cannot be simulated or tested on historical data. Also, since financial markets keep evolving fast, the future may exhibit patterns not present in historical data. However, if a strategy cannot prove itself valid in a backtest most probably will never work in real trading. Backtesting can at least help us to weed out the strategies that do not prove themselves worthy.

Several frameworks make it easy to backtest trading strategies using Python. Two popular examples are Zipline and Backtrader. Frameworks like Zipline and Backtrader include all the tools needed to design, test, and implement an algorithmic trading strategy. They can even automate the submission of real orders to an execution broker.

With this article, we are taking a different approach: we want to investigate how to build and test a trading system from scratch using Python, pandas, and NumPy as our only tools. Why should we want to do that? To start with, building a backtest from scratch is an excellent exercise and will help to understand in detail how a strategy works. Also, we may find ourselves in situations where we need to implement solutions not available in existing frameworks. Or, you may want to embark on the journey of creating your own backtesting framework!

## Backtesting our first system

We can create a barebone backtest using Python with NumPy and pandas. To build an example, we are going to use prices for the Campbell Soup Company stock (CPB) traded on the NYSE. I downloaded five years of trading history from Yahoo! Finance: the file is available here.

We start by setting up our environment and load the price series into a data frame:

In [1]:
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
pd.plotting.register_matplotlib_converters()

# This is needed if you're using Jupyter to visualize charts:
%matplotlib inline

import sys
import matplotlib as mpl
import mplfinance as mpf
print(f"Python version: {sys.version}")
print(f"pandas version: {pd.__version__}")
print(f"numpy version: {np.__version__}")
print(f"mplfinanceversion: {mpf.__version__}")

datafile = 'data/CPB.csv'
data = pd.read_csv(datafile, index_col = 'Date')
data.index = pd.to_datetime(data.index) # Converting the dates from string to datetime format

data

Out [1]:
Python version: 3.7.7 (default, May  6 2020, 04:59:01)
[Clang 4.0.1 (tags/RELEASE_401/final)]
pandas version: 1.1.0
numpy version: 1.19.1
mplfinanceversion: 0.12.7a0

Open High Low Close Adj Close Volume
Date
2015-09-02 47.389999 47.770000 47.009998 47.770000 41.054787 2547200
2015-09-03 46.910000 48.680000 46.070000 48.529999 41.707958 2364100
2015-09-04 48.230000 48.500000 47.439999 47.939999 41.200890 2019300
2015-09-08 48.500000 49.410000 48.500000 49.380001 42.438469 2458700
2015-09-09 49.779999 49.779999 48.590000 48.720001 41.871243 2198900
... ... ... ... ... ... ...
2020-08-26 52.900002 53.500000 52.389999 53.480000 53.480000 1375400
2020-08-27 53.509998 54.080002 53.259998 53.290001 53.290001 1432100
2020-08-28 53.290001 53.290001 51.930000 52.139999 52.139999 1827300
2020-08-31 52.349998 52.709999 52.020000 52.610001 52.610001 2018800
2020-09-01 52.810001 52.880001 51.189999 51.400002 51.400002 3060900

1259 rows × 6 columns

### A basic strategy

For our example, we are going to test a basic moving average crossover system based on a 20-day Exponential Moving Average (EMA) and a 200-day Simple Moving Average (SMA) of the daily closing price (using Adjusted Close in this example). We are going to buy the stock (take a long position) whenever the 20-day EMA crosses the 200-day SMA from below.

We add columns with our moving averages to the data frame:

In [2]:
df = data.copy()

sma_span = 200
ema_span = 20

df.round(3)

Out [2]:
Open High Low Close Adj Close Volume sma200 ema20
Date
2015-09-02 47.39 47.77 47.01 47.77 41.055 2547200 NaN 41.055
2015-09-03 46.91 48.68 46.07 48.53 41.708 2364100 NaN 41.398
2015-09-04 48.23 48.50 47.44 47.94 41.201 2019300 NaN 41.325
2015-09-08 48.50 49.41 48.50 49.38 42.438 2458700 NaN 41.647
2015-09-09 49.78 49.78 48.59 48.72 41.871 2198900 NaN 41.701
... ... ... ... ... ... ... ... ...
2020-08-26 52.90 53.50 52.39 53.48 53.480 1375400 48.478 51.571
2020-08-27 53.51 54.08 53.26 53.29 53.290 1432100 48.519 51.735
2020-08-28 53.29 53.29 51.93 52.14 52.140 1827300 48.553 51.773
2020-08-31 52.35 52.71 52.02 52.61 52.610 2018800 48.584 51.853
2020-09-01 52.81 52.88 51.19 51.40 51.400 3060900 48.611 51.810

1259 rows × 8 columns

As we can see, by using a 200-day SMA we get NaN values in the first 199 rows for the respective column. This is just an exercise, and we can just get rid of those rows to perform our backtest. In real practice, we may consider using a different indicator to avoid losing so much data. To remove the NaN values:

In [3]:
df.dropna(inplace=True)

df.round(3)

Out [3]:
Open High Low Close Adj Close Volume sma200 ema20
Date
2016-06-17 62.75 62.75 61.87 62.44 54.582 2064300 49.113 54.320
2016-06-20 62.63 63.15 62.39 62.41 54.556 1459400 49.181 54.343
2016-06-21 62.65 63.21 62.49 62.85 54.940 1161900 49.247 54.400
2016-06-22 63.08 63.08 62.23 62.59 54.713 1395800 49.314 54.429
2016-06-23 62.68 62.92 62.27 62.64 54.757 1177000 49.376 54.461
... ... ... ... ... ... ... ... ...
2020-08-26 52.90 53.50 52.39 53.48 53.480 1375400 48.478 51.571
2020-08-27 53.51 54.08 53.26 53.29 53.290 1432100 48.519 51.735
2020-08-28 53.29 53.29 51.93 52.14 52.140 1827300 48.553 51.773
2020-08-31 52.35 52.71 52.02 52.61 52.610 2018800 48.584 51.853
2020-09-01 52.81 52.88 51.19 51.40 51.400 3060900 48.611 51.810

1060 rows × 8 columns

Let’s have a look at our data in a chart:

In [4]:
def plot_system1(data):
df = data.copy()
dates = df.index
sma200 = df['sma200']
ema20 = df['ema20']

with plt.style.context('fivethirtyeight'):
fig = plt.figure(figsize=(14,7))
plt.plot(dates, price, linewidth=1.5, label='CPB price - Daily Adj Close')
plt.plot(dates, sma200, linewidth=2, label='200 SMA')
plt.plot(dates, ema20, linewidth=2, label='20 EMA')
plt.title("A Simple Crossover System")
plt.ylabel('Price($)') plt.legend() plt.show() # This is needed only if not in Jupyter plot_system1(df)  Out [4]: To keep track of our positions in the data frame we add a column that contains, for each row, the number 1 for the days when we have a long position and the number 0 for the days when we have no position: In [5]: # Our trading condition: long_positions = np.where(df['ema20'] > df['sma200'], 1, 0) df['Position'] = long_positions df.round(3)  Out [5]: Open High Low Close Adj Close Volume sma200 ema20 Position Date 2016-06-17 62.75 62.75 61.87 62.44 54.582 2064300 49.113 54.320 1 2016-06-20 62.63 63.15 62.39 62.41 54.556 1459400 49.181 54.343 1 2016-06-21 62.65 63.21 62.49 62.85 54.940 1161900 49.247 54.400 1 2016-06-22 63.08 63.08 62.23 62.59 54.713 1395800 49.314 54.429 1 2016-06-23 62.68 62.92 62.27 62.64 54.757 1177000 49.376 54.461 1 ... ... ... ... ... ... ... ... ... ... 2020-08-26 52.90 53.50 52.39 53.48 53.480 1375400 48.478 51.571 1 2020-08-27 53.51 54.08 53.26 53.29 53.290 1432100 48.519 51.735 1 2020-08-28 53.29 53.29 51.93 52.14 52.140 1827300 48.553 51.773 1 2020-08-31 52.35 52.71 52.02 52.61 52.610 2018800 48.584 51.853 1 2020-09-01 52.81 52.88 51.19 51.40 51.400 3060900 48.611 51.810 1 1060 rows × 9 columns Whatever rule we are trying to implement, it’s a good idea to inspect our signals to make sure that everything works as intended. Pesky mistakes like to hide inside this kind of computations: it’s way too easy to start testing a system, only to discover later that we did not implement our rules correctly. In particular, we need to be wary of introducing any form of look-ahead bias: this happens when we include in our trading rule data that is not actually available at the time the rule is evaluated. If a system backtest produces results that are too good to be true, look-ahead bias is the most likely culprit. We can inspect our signals by checking the numeric variables in our data frame and by plotting the signals on a chart. To select the days when the trading signal is triggered: In [6]: buy_signals = (df['Position'] == 1) & (df['Position'].shift(1) == 0) df.loc[buy_signals].round(3)  Out [6]: Open High Low Close Adj Close Volume sma200 ema20 Position Date 2016-12-30 60.93 61.09 60.33 60.47 53.802 1204500 52.734 52.833 1 2017-05-25 58.45 58.86 57.65 58.83 52.975 1858600 51.651 51.778 1 2019-04-10 39.19 39.57 38.91 39.13 37.678 2936200 35.912 36.021 1 We have only three signals in this case. To make sure that we are applying the crossover trading rule correctly, we can include the days before the signals in the selection: In [7]: buy_signals_prev = (df['Position'].shift(-1) == 1) & (df['Position'] == 0) df.loc[buy_signals | buy_signals_prev].round(3)  Out [7]: Open High Low Close Adj Close Volume sma200 ema20 Position Date 2016-12-29 60.56 61.02 60.48 60.95 54.229 940200 52.745 52.731 0 2016-12-30 60.93 61.09 60.33 60.47 53.802 1204500 52.734 52.833 1 2017-05-24 58.16 58.43 57.93 58.38 52.570 2204000 51.655 51.652 0 2017-05-25 58.45 58.86 57.65 58.83 52.975 1858600 51.651 51.778 1 2019-04-09 38.46 39.81 38.46 39.49 37.688 5173300 35.903 35.846 0 2019-04-10 39.19 39.57 38.91 39.13 37.678 2936200 35.912 36.021 1 Everything looks good so far: on the days before the signal, ema20 is below sma200, and it crosses above on the signal days. We can do a similar inspection for the signals to exit our positions: I leave this exercise to you. We can plot markers for our signals on a chart: In [8]: def plot_system1_sig(data): df = data.copy() dates = df.index price = df['Adj Close'] sma200 = df['sma200'] ema20 = df['ema20'] buy_signals = (df['Position'] == 1) & (df['Position'].shift(1) == 0) buy_marker = sma200 * buy_signals - (sma200.max()*.05) buy_marker = buy_marker[buy_signals] buy_dates = df.index[buy_signals] sell_signals = (df['Position'] == 0) & (df['Position'].shift(1) == 1) sell_marker = sma200 * sell_signals + (sma200.max()*.05) sell_marker = sell_marker[sell_signals] sell_dates = df.index[sell_signals] with plt.style.context('fivethirtyeight'): fig = plt.figure(figsize=(14,7)) plt.plot(dates, price, linewidth=1.5, label='CPB price - Daily Adj Close') plt.plot(dates, sma200, linewidth=2, label='200 SMA') plt.plot(dates, ema20, linewidth=2, label='20 EMA') plt.scatter(buy_dates, buy_marker, marker='^', color='green', s=160, label='Buy') plt.scatter(sell_dates, sell_marker, marker='v', color='red', s=160, label='Sell') plt.title("A Simple Crossover System with Signals") plt.ylabel('Price($)')
plt.legend()

plt.show() # This is needed only if not in Jupyter

plot_system1_sig(df)

Out [8]:

From the chart, we can see that there is a mismatch between Buy and Sell signals. The first signal is a Sell (without a Buy) since we start from a long position at the beginning of the series. The last signal is a Buy (without a Sell) because we keep a long position at the end of the series.

### Strategy returns

We can now compute the return for our strategy as a percentage of the initial investment and compare it to the returns of a Buy and Hold strategy that simply buys our stock at the beginning of the period and holds it until the end.

The price series that we are going to use to calculate the returns is Adjusted Close: by using an adjusted price we make sure that the effects of dividends, stock splits, and other corporate actions on returns are taken into account in our calculation.

In [9]:
# The returns of the Buy and Hold strategy:

# The returns of the Moving Average strategy:
df['Strategy'] = df['Position'].shift(1) * df['Hold']

# We need to get rid of the NaN generated in the first row:
df.dropna(inplace=True)

df

Out [9]:
Open High Low Close Adj Close Volume sma200 ema20 Position Hold Strategy
Date
2016-06-20 62.630001 63.150002 62.389999 62.410000 54.555550 1459400 49.180615 54.342710 1 -0.000481 -0.000481
2016-06-21 62.650002 63.209999 62.490002 62.849998 54.940170 1161900 49.246776 54.399611 1 0.007025 0.007025
2016-06-22 63.080002 63.080002 62.230000 62.590000 54.712894 1395800 49.314336 54.429448 1 -0.004145 -0.004145
2016-06-23 62.680000 62.919998 62.270000 62.639999 54.756611 1177000 49.375926 54.460606 1 0.000799 0.000799
2016-06-24 61.450001 63.490002 61.049999 62.400002 54.546814 3845400 49.439304 54.468816 1 -0.003839 -0.003839
... ... ... ... ... ... ... ... ... ... ... ...
2020-08-26 52.900002 53.500000 52.389999 53.480000 53.480000 1375400 48.478148 51.571117 1 0.008450 0.008450
2020-08-27 53.509998 54.080002 53.259998 53.290001 53.290001 1432100 48.519216 51.734820 1 -0.003559 -0.003559
2020-08-28 53.290001 53.290001 51.930000 52.139999 52.139999 1827300 48.553018 51.773409 1 -0.021816 -0.021816
2020-08-31 52.349998 52.709999 52.020000 52.610001 52.610001 2018800 48.584327 51.853084 1 0.008974 0.008974
2020-09-01 52.810001 52.880001 51.189999 51.400002 51.400002 3060900 48.611054 51.809934 1 -0.023268 -0.023268

1059 rows × 11 columns

The returns for the whole period are simply a sum of the daily log returns:

In [10]:
returns = np.exp(df[['Hold', 'Strategy']].sum()) - 1

print(f"Strategy return: {round(returns['Strategy']*100,2)}%")

Out [10]:
Buy and hold return: -5.83%
Strategy return: 10.3%


Those returns are related to a period of 1060 days. If we want to compare them to returns from other periods we need to annualize them:

In [11]:
n_days = len(df)

# Assuming 252 trading days in a year:
ann_returns = 252 / n_days * returns

print(f"Buy and hold annualized return: {round(ann_returns['Hold']*100,2)}%")
print(f"Strategy annualized return: {round(ann_returns['Strategy']*100,2)}%")

Out [11]:
Buy and hold annualized return: -1.39%
Strategy annualized return: 2.45%


So far, our simple strategy seems to do better than to buy and hold: this particular stock had a negative trend over the period, while the simple MA crossover strategy gave us a return in excess of 10%. I have actually chosen that stock series on purpose: it has a bearish trend at the beginning of the period before turning bullish. Do not expect to be always so lucky with the results!

Unless you are familiar with log returns, you may wonder why and how we used logarithms in the return calculations. Here is a bit of maths to explain that in case that sounds all new to you. Otherwise, feel free to skip to the next section.

In quantitative finance it’s very common to use logarithms to calculate returns: they make some computations easier to handle. If the daily return $R_t$ is defined as:

$R_t = \frac{P_t - P_{t-1}}{P_{t-1}}$

where $P_t$ is the price at date $t$, the logarithmic return $r_t$ is defined as:

$r_t = \ln (1 + R_t)$

By applying some basic algebra, it’s possible to compute the daily log return as:

$r_t = \ln \frac{P_t}{P_{t-1}} = \ln P_t - \ln P_{t-1}$

Why are log-returns handy? If we have a series of daily returns and we need to calculate the return for the whole period, with log returns we can make a sum of them By contrast, with regular returns, we need to a multiplication:

$\text{Return over the period} = (1+R_1)\times(1+R_2)\times\dots\times(1+R_T) = \exp \left( \sum_{t=0}^{T} r_t \right)$

where T is the number of days in the period of time that we are considering. Calculating the annualized returns comes easier as well.

## A more complex strategy

The strategy we have just tested had only two possible positions: we were either long (holding the stock) or flat (holding just cash). It would be interesting to try and test a strategy that adds the possibility of a short position as well (selling borrowed shares and buying them back when exiting the position). To build an example of this strategy we include two simple moving averages, one for the daily highs, and one for the daily lows. We also add a 15-day exponential moving average. We take positions based on the following rules:

• when the EMA is above the higher SMA (plus a 2% offset), we take a long position (buy)
• when the EMA is below the lower SMA (minus a 2% offset), we take a short position (sell short)
• in all other situations (EMA between the to SMAs), we keep out of the market

I added the offset to the SMAs to reduce the numbers of false signals. Let’s prepare a new data frame:

In [12]:
df2 = data.copy()

sma_span = 40
ema_span = 15

df2['H_sma'] = df2['High'].rolling(sma_span).mean()
df2['L_sma'] = df2['Low'].rolling(sma_span).mean()
df2['C_ema'] = df2['Close'].ewm(span=ema_span).mean()

df2.dropna(inplace=True)

df2.round(3)

Out [12]:
Open High Low Close Adj Close Volume H_sma L_sma C_ema
Date
2015-10-28 51.27 51.38 50.33 50.79 43.917 1368900 50.745 49.844 50.660
2015-10-29 50.69 51.37 50.38 51.20 44.272 1035700 50.835 49.928 50.728
2015-10-30 51.11 51.45 50.78 50.79 43.917 1184300 50.904 50.046 50.735
2015-11-02 50.88 50.99 50.36 50.74 43.874 1169700 50.966 50.119 50.736
2015-11-03 50.50 50.75 49.70 50.23 43.433 1278200 51.000 50.149 50.673
... ... ... ... ... ... ... ... ... ...
2020-08-26 52.90 53.50 52.39 53.48 53.480 1375400 50.791 49.927 51.923
2020-08-27 53.51 54.08 53.26 53.29 53.290 1432100 50.897 50.029 52.094
2020-08-28 53.29 53.29 51.93 52.14 52.140 1827300 50.983 50.100 52.100
2020-08-31 52.35 52.71 52.02 52.61 52.610 2018800 51.054 50.175 52.163
2020-09-01 52.81 52.88 51.19 51.40 51.400 3060900 51.130 50.229 52.068

1220 rows × 9 columns

Here we are making use of High and Low prices in addition to Close prices. To plot those values on a chart, it’s a good idea to use OHLC bars or candlesticks. We are going to use the mplfinance library for this. If you have not done that yet, you can easily install mplfinance using:

pip install --upgrade mplfinance

To integrate the candlestick chart with our existing style, I am going to apply the External Axes Method of mplfinance:

In [13]:
import mplfinance as mpf
import numpy as np

def plot_system2(data):
df2 = data.copy()
dates = np.arange(len(df2)) # We need this for mpl.plot()
h_sma = df2['H_sma']*1.02
l_sma = df2['L_sma']*.98
c_ema = df2['C_ema']

with plt.style.context('fivethirtyeight'):
fig = plt.figure(figsize=(14,7))
ax = plt.subplot(1,1,1)
ax.plot(dates, h_sma, linewidth=2, color='blue', label='High 40 SMA + 2%')
ax.plot(dates, l_sma, linewidth=2, color='blue', label='Low 40 SMA - 2%')
ax.plot(dates, c_ema, linewidth=1.5, color='red', linestyle='--', label='Close 15 EMA')
plt.title("A System with Long-Short Positions")
ax.set_ylabel('Price($)') plt.legend() plt.show() # This is needed outside of Jupyter plot_system2(df2)  Out [13]: We can examine more in detail any specific range of dates: In [14]: plot_system2(df2['2019-07-01':'2019-12-31'])  Out [14]: We then apply our trading rule and add the position columns: In [15]: offset = 0.02 long_positions = np.where(df2['C_ema'] > df2['H_sma']*(1+offset), 1, 0) short_positions = np.where(df2['C_ema'] < df2['L_sma']*(1-offset), -1, 0) df2['Position'] = long_positions + short_positions df2.round(3)  Out [15]: Open High Low Close Adj Close Volume H_sma L_sma C_ema Position Date 2015-10-28 51.27 51.38 50.33 50.79 43.917 1368900 50.745 49.844 50.660 0 2015-10-29 50.69 51.37 50.38 51.20 44.272 1035700 50.835 49.928 50.728 0 2015-10-30 51.11 51.45 50.78 50.79 43.917 1184300 50.904 50.046 50.735 0 2015-11-02 50.88 50.99 50.36 50.74 43.874 1169700 50.966 50.119 50.736 0 2015-11-03 50.50 50.75 49.70 50.23 43.433 1278200 51.000 50.149 50.673 0 ... ... ... ... ... ... ... ... ... ... ... 2020-08-26 52.90 53.50 52.39 53.48 53.480 1375400 50.791 49.927 51.923 1 2020-08-27 53.51 54.08 53.26 53.29 53.290 1432100 50.897 50.029 52.094 1 2020-08-28 53.29 53.29 51.93 52.14 52.140 1827300 50.983 50.100 52.100 1 2020-08-31 52.35 52.71 52.02 52.61 52.610 2018800 51.054 50.175 52.163 1 2020-09-01 52.81 52.88 51.19 51.40 51.400 3060900 51.130 50.229 52.068 0 1220 rows × 10 columns We can plot our signals on the chart: In [16]: def plot_system2_sig(data): df2 = data.copy() dates = np.arange(len(df2)) # We need this for mpl.plot() price = df2['Adj Close'] h_sma = df2['H_sma']*1.02 l_sma = df2['L_sma']*.98 c_ema = df2['C_ema'] def reindex_signals(signals, markers): ''' - takes two pd.Series (boolean, float) - returns signals and markers reindexed to an integer range, and their index ''' signals.reset_index(drop=True, inplace=True) signals = signals[signals==True] dates = signals.index markers = markers[dates] markers.index = dates return signals, markers, dates buy_signals = (df2['Position'] == 1) & (df2['Position'].shift(1) != 1) buy_marker = h_sma * buy_signals[buy_signals==True] - (h_sma.max()*.04) buy_signals, buy_marker, buy_dates = reindex_signals(buy_signals, buy_marker) exit_buy_signals = (df2['Position'] != 1) & (df2['Position'].shift(1) == 1) exit_buy_marker = h_sma * exit_buy_signals + (h_sma.max()*.04) exit_buy_signals, exit_buy_marker, exit_buy_dates = reindex_signals(exit_buy_signals, exit_buy_marker) sell_signals = (df2['Position'] == -1) & (df2['Position'].shift(1) != -1) sell_marker = l_sma * sell_signals + (l_sma.max()*.04) sell_signals, sell_marker, sell_dates = reindex_signals(sell_signals, sell_marker) exit_sell_signals = (df2['Position'] != -1) & (df2['Position'].shift(1) == -1) exit_sell_marker = l_sma * exit_sell_signals - (l_sma.max()*.04) exit_sell_signals, exit_sell_marker, exit_sell_dates = reindex_signals(exit_sell_signals, exit_sell_marker) with plt.style.context('fivethirtyeight'): fig = plt.figure(figsize=(14,7)) ax = plt.subplot(1,1,1) mpf.plot(df2, ax=ax, show_nontrading=False, type='candle') ax.plot(dates, h_sma, linewidth=2, color='blue', label='High 40 SMA + 2%') ax.plot(dates, l_sma, linewidth=2, color='blue', label='Low 40 SMA - 2%') ax.plot(dates, c_ema, linewidth=1.5, color='red', linestyle='--', label='Close 15 EMA') ax.scatter(buy_dates, buy_marker, marker='^', color='green', s=160, label='Buy') ax.scatter(exit_buy_dates, exit_buy_marker, marker='v', s=160, label='Exit Buy') ax.scatter(sell_dates, sell_marker, marker='v', color='red', s=160, label='Sell') ax.scatter(exit_sell_dates, exit_sell_marker, marker='^', color='orange', s=160, label='Exit Sell') plt.title("A System with Long-Short Signals") ax.set_ylabel('Price($)')
plt.legend()

plt.show() # This is needed outside of Jupyter

plot_system2_sig(df2)

Out [16]:

This system has more signals than the previous one and the chart looks quite crowded. We can take a look at any date range in detail:

In [17]:
plot_system2_sig(df2['2018-12-01':'2019-05-30'])

Out [17]:

We apply the same calculation as before to obtain the strategy’s return:

In [18]:
# The returns of the Buy and Hold strategy:

# The returns of the Moving Average strategy:
df2['Strategy'] = df2['Position'].shift(1) * df2['Hold']

# We need to get rid again of the NaN generated in the first row:
df2.dropna(inplace=True)

returns2 = np.exp(df2[['Hold', 'Strategy']].sum()) -1

print(f"Strategy return: {round(returns2['Strategy']*100,2)}%")

Out [18]:
Buy and hold return: 17.04%
Strategy return: -5.25%


As before, we can annualize the returns:

In [19]:
n_days2 = len(df2)

# Assuming 252 trading days in a year:
ann_returns2 = 252 / n_days2 * returns2

print(f"Buy and hold annualized return: {round(ann_returns2['Hold']*100,2)}%")
print(f"Strategy annualized return: {round(ann_returns2['Strategy']*100,2)}%")

Out [19]:
Buy and hold annualized return: 3.52%
Strategy annualized return: -1.09%


In this case, our strategy is actually worse better than the buy and hold strategy.

You might have noted that I used the unadjusted price series to evaluate the signals, while I kept using adjusted prices to calculate returns. Evaluating signals using non-adjusted prices has the risk to introduce false triggers whenever dividends, splits, or other corporate actions create a gap in the prices. Here, I just used a price series that is commonly available and that everyone can download for free. If all we have is unadjusted prices, we should correct our backtest using all the information available about corporate actions.

## Conclusion

Is that all we need to perform backtests and to select strategies we can rely on? Definitely not: in our backtests, we made (although implicitly) some assumptions and simplifications that can substantially affect our results. To start with, we assumed that a stock can be bought at exactly the closing price of the day the signal is triggered. This is, in reality, not guaranteed: the actual price will be somewhere in the range of the day after the signal occurred. Then, transaction costs have to be included. For example:

• Brokerage fees are paid to execute and clear our orders.
• The spread between Bid and Ask price is a cost component.
• If we buy on leverage we need to pay interest. Similarly, if we borrow shares to sell short we need to pay interest on that loan.

Some of those factors are more evident than others to understand and include in the model.

When we want to evaluate the performance of a system and compare it to that of other systems, the return over a given period is only one of the many performance and risk metrics that we want to consider. Some examples are: